The paper deals with the formulation of a consistent set of governing equations apt to describe the physical phenomenology comprising the hypersonic flow field of an ionized gas mixture and the electromagnetic field. The governing equations of the flow field and those of the electromagnetic field are revisited in sequence and differences or similarities with past treatments are pointed out and discussed. The equations governing the flow field hinge on the customary balance of masses, momenta and energies. The equations governing the electromagnetic field are introduced both directly in terms of the Maxwell equations and by recourse to the scalar and vector potentials. The theory of linear irreversible thermodynamics based on the entropy-balance equation is also revisited for the purpose of obtaining, consistently with the presence of the electromagnetic field, the phenomenological relations required to bring the governing equations into a mathematically closed form. Old problems, such as the influence of the medium compressibility on chemical-relaxation rates or the importance of cross effects among generalized fluxes and forces, are re-discussed; additional problems, such as the necessity to consider the tensorial nature of the transport properties because of the presence of the magnetic field, are pointed out. A non-conventional choice of first-tensorial-order generalized forces and corresponding fluxes is proposed which appears to offer more simplicity and better convenience from a conceptual point of view when compared to alternative definitions customarily used in the literature. The applicability domain of the present formulation is clearly outlined and recommendations for further work are given.